# Working Group 2021/2022

With Jacques Féjoz and Jean-Pierre Marco, we organize a working group on Hamiltonian and symplectic dynamics. It takes place once per week at Université Dauphine-PSL or Jussieu.

15 June 2022 (Jussieu) : Alexey Glutsyuk

8 June 2022 (Dauphine) : Alain Chenciner

Title : ABC

Abstract : Limiter à trois la dimension de l’espace dans lequel on étudie le problème newtonien des n corps peut sembler naturel d’un point de vue physique mais diverses raisons, en particulier la structure algébrique des équations, invitent le mathématicien à s’affranchir de ce qui apparaît comme une contrainte. C’est ainsi que nous rencontrerons sous la forme de matrices les trois lettres formant le titre – A comme forces, B comme forme, C comme moment cinétique – en étudiant les configurations équilibrées de n corps, configurations généralisant les classiques configurations centrales en ce que, soumises à une attraction de type newtonien, elles admettent un mouvement d’équilibre relatif, en général quasi-périodique, dans un espace dont la dimension peut dépasser trois.

18 May 2022 (Jussieu, salle 15-25-502) : Boris Khesin

Title : Hamiltonian geometry behind compressible fluids

Abstract : We compare Arnold’s geodesic setting for incompressible
fluids and a geometric framework for Newton’s equations on
diffeomorphism groups and smooth probability densities. It turns out
that several important PDEs of hydrodynamical origin can be described
in this framework in a natural way. In particular, the Madelung
transform between the Schrödinger-type equations on wave functions and
Newton’s equations on densities turns out to be a Kähler map between
the corresponding phase spaces, equipped with the Fubini-Study and
Fisher-Rao information metrics. This is a joint work with G.Misiolek
and K.Modin.

16 February 2022 (Jussieu, salle 15-16-411) : Michiel Burgelman

Title : Subharmonic and chaotic dynamics in driven Josephson circuits

Abstract : One of the leading platforms for building quantum technologies are superconducting circuits. Here, a promising approach for storing and manipulating quantum information consists of coupling linear resonators carrying the quantum information through nonlinear couplers mediating interactions. The ubiquitous nonlinear element of choice is the Josephson junction, which is combined with linear circuit elements to design different nonlinear couplers. Desired interactions are then activated parametrically, by applying drives to the junction at well-chosen frequencies. The interaction rate can then be enhanced by increasing the amplitude of the driving. Recent superconducting circuits experiments have shown limitations when driving strongly, with unexpected changes in the system response when ramping up the driving amplitude. To understand the origin of these problems, there has been an effort to obtain a deeper understanding of the dynamics of the junction itself (2D system). The driven Josephson junction is well-known to  display a vast variety of possible nonlinear phenomena, the most elementary of which are the DC- and AC-Josephson effect, but various subharmonic and chaotic regimes have also been reported. The main contribution of our work is twofold. On the one hand, we identify the occurrence of chaotic junction dynamics as a limiting factor for typical superconducting circuits experiments involving periodic drives. We do this by studying the quantum signatures of classical chaos via a numerical approach based on Floquet theory. On the other hand, we show how to choose the junction parameters to effectively suppress this chaos, while still retaining the desirable resonant (subharmonic) behavior of the junction. We will begin by defining the classical model of the periodically driven junction, and the Poincaré map associated to it.  After presenting a preliminary numerical account of the Poincaré map upon changing parameters, we identify one main parameter that determines the chaotic behavior of the system. We show that by making the system more linear, we can seemingly suppress the chaotic dynamics altogether. In the non-chaotic regime, treating the junction as a perturbation of a linear system, we study subharmonic solutions of fixed order, through a first-order average of the system around resonance.  Next, the quantum signature of classical chaos is explained, including a prior on quantum mechanics, and a systematic numerical study of chaotic behavior is performed around the 3:1-resonance of the system. We conclude with a more informal discussion of some open questions, including the possible routes to chaos, with regard to a conjecture of Tresser concerning period-doubling cascades.

26 January 2022 (Jussieu, salle 15-16-411) : Jacques Féjoz

Title : Sur les sous-variétés invariantes en dynamique conforme symplectique – isotropie et entropie

Abstract : Les systèmes dynamiques conformes symplectiques sont une extension des systèmes dynamiques symplectiques, où la forme symplectique peut être modifiée proportionnellement à elle-même. Ils incluent les systèmes
mécaniques où la force de frottement est parallèle à la vitesse. Pour les systèmes hamiltoniens, Herman avait remarqué que les tores
invariants quasipériodiques minimaux (comme ceux de la théorie KAM) sont
forcément isotropes. Nous généraliserons ce résultat au cas conforme
symplectique, en reliant l’isotropie à l’entropie de la dynamique
restreinte. Un outil clef est l’inégalité de Yomdin, ainsi que
l’amélioration obtenue en analysant l’effet des entropies locales,
transversalement au feuilletage caractéristique de la sous-variété
invariante. Travail avec Marie-Claude Arnaud.

19 January 2022 (online) : Santiago Barbieri

Title : On the genericity of effectively stable integrable Hamiltonian systems and on their algebraic properties (Part II) (Joint work with L. Niederman)

Abstract : In this second part, I will give further elements of proof of the results that were presented in the first seminar about the genericity of the steepness property. Namely, I will start by describing an algebraic property that characterizes the gradient of non-steep polynomials and which Nekhoroshev calls “s-vanishing condition”, where s is any positive integer greater or equal than two. Requesting the s-vanishing condition for a polynomial amounts to asking that the derivatives up to order s of the restriction of its gradient to the uniform curve of minima described in the previous seminar are all equal to zero. Two important results arise at this point: 1) The fact that the minimal curve is parametrized by one of the coordinates implies that the set of s-vanishing polynomials in the space of r-jets (with r strictly greater than s) has codimension greater or equal than s. Moreover, this set turns out also to be semi-algebraic. 2) Moreover, the study of the s-vanishing condition – together with the Bernstein inequality satisfied by the minimal curve – yields that all polynomials belonging to any open neighborhood outside of the set of s-vanishing polynomials satisfy a uniform lower estimate on the norm of their gradient. These two bricks, together with a well known result on the variation of smooth functions, imply the genericity of steepness: for r a sufficiently high positive integer, the set of non-steep polynomials in the space of r-jets is contained in the set of s-vanishing polynomials for s equal to the integer part of r/2. Moreover, an attentive study of the equations and inequalities defining the set of s-vanishing polynomials allows to find a recursion algorithm which yields its explicit form for any order r of the space of jets. This last point, in particular, proves fundamental in order to give a sufficient criterion to establish whether a given function is steep or not (an important point in view of applications).

12 January 2022 (online) : Santiago Barbieri

Title : On the genericity of effectively stable integrable Hamiltonian systems and on their algebraic properties (Part I) (Joint work with L. Niederman)

Abstract : Hamiltonian systems constitute an important class of dynamical systems. Those hamiltonian systems which are integrable in the sense of Arnold-Liouville possess an important property: their solutions can be written explicitly and the phase space is foliated by invariant tori carrying global quasi-periodic orbits. This kind of systems are exceptional but in applications it is not rare to see systems which are perturbations of integrable ones. A natural question is then to determine whether the stability of solutions is preserved for this latter type of systems. Kolmogorov-Arnold-Moser theory assures that, under generic hypotheses, a Cantor set of positive Lebesgue measure of invariant tori carrying quasi-periodic motions persists under a sufficiently small perturbation. On the other hand, instabilities may appear in the complementary of this set (Arnold diffusion). Moreover, a theorem due to Nekhoroshev (1971-1977) shows that the solutions of a sufficiently regular integrable system verifying a transversality property on its gradient – known as “steepness” – are stable over a long time under the effect of a suitably small perturbation. Nekhoroshev also showed (1973) that the steepness property is generic, both in measure and topologic sense, in the space of jets (Taylor polynomials) of sufficiently smooth functions. However, the proof of this result kept being poorly understood up to now and, surprisingly, the paper in which it is contained is hardly known, whereas the rest of the theory has been widely studied over the decades. Moreover, the definition of steepness is not constructive and no general rule to establish whether a given function is steep or not existed up to now, thus entailing a major problem in applications.
In this first seminar, I will start by explaining the main ideas behind Nekhoroshev’s proof of the genericity of steepness by making use of a more modern language. Indeed, the proof strongly relies on arguments of complex analysis and real algebraic geometry: the latter was much less developed than nowadays at the time that Nekhoroshev was writing, so that many passages appear to be quite obscure in the original article. In particular, I will show that the first step in order to prove the genericity of steepness consists in constructing an analytic curve of minimal points for the norm of the gradient of the Taylor polynomial of any sufficiently regular function. It turns out that such a curve can always be constructed and has the following properties
1) it is parametrized by one of the coordinates;
2) it is defined over a real interval whose length only depends on the degree of the Taylor polynomial that one is considering (thus, it does not depend on the function);
3) similarly, its complex analyticity width only depends on the degree of the Taylor polynomial that one is considering;
4) there exists a pure constant C such that the size of each component of the curve in the complex domain is bounded by the product of C times the length of the real interval of definition. Point 4), in particular, is a consequence of a type of Bernstein inequality for semialgebraic functions that (up to our knowledge) seems to have been proved by Nekhoroshev for the first time and to have been rediscovered (by exploiting different techniques) by Roytwarf and Yomdin in the 1990s. In the end of the seminar, I will take a little detour to explain Nekhoroshev’s strategy to prove the Bernstein inequality and I will show a possible improvement of this result that we have found and which we ignore if it is already known or not.

Reference: N. N. Nekhoroshev, “Stable lower estimates for smooth mappings and for gradients of smooth function”, Mathematics of the USSR-Sbornik, 1973, vol. 90 (132), no. 3, pp.432-478.

8 December 2021 (Dauphine, salle A711) : Paul Ramond

Title : Archimèdes, Kepler, Birkhoff et Hénon en Isochronie

Abstract : Les potentiels isochrones ont été introduits par Michel Hénon à la fin des années 50. Ce sont des potentiels de R^3 dans R, sphériques, i.e., qui ne dépendent que de la distance au centre r. Une particule orbitant dans un tel potentiel a, de façon générale, deux intégrales premières: l’énergie et le moment cinétique. Les potentiels isochrones ont la particularité (qui les défini) de ne contenir que des orbites qui, quand elles sont bornées, ont une période radiale qui ne dépend pas du moment cinétique de la particule. Les plus célèbres sont le potentiel harmonique (ou Hooke) en r^2, dont la période est une constante pure, et le potentiel de Kepler (ou Newton) en -1/r, dont la période ne dépend que de l’énergie. Il en existe d’autres, comme l’a montré Hénon, mais il ne les a pas étudié en profondeur. Je présenterai mon travail de thèse dans ce cadre, qui consiste en l’étude poussée et aussi complète que possible, de ces potentiels isochrones, de leur structure et des orbites qu’ils contiennent. On verra en particulier : (i) qu’il existe 5 grandes familles d’isochrones, (ii) que toutes les orbites vérifient l’équation de Kepler Ω t = E – e sin E et la loi de Kepler T^2 = cst. E^-3, (iii) que tout potentiel isochrone est en relation univoque avec une parabole du plan par un changement de variable simple, (iv) que le théorème de Bertrand est un corollaire des propriétés générales de l’isochronie, et bien d’autres propriétés remarquables. Je présenterai des démonstrations purement géométriques (au sens euclidien ou symplectique du terme), et ce faisant nous parlerons d’Archimedes, de Kepler, de Huygens, de Birkhoff et de Hénon.

1 December 2021 (Dauphine, salle A711) :

24 November 2021 (Dauphine, salle A711) : Skander Charfi

17 November 2021 : no meetings

10 November 2021 (Dauphine, salle A711) : Flavien Grycan-Gerard

Title : Entropie polynomiale des flots hamiltoniens à singularités
modérées sur les surfaces symplectiques

Abstract : Les systèmes complètement intégrables ont en général une
dynamique simple : pour la plupart d’entre eux l’entropie topologique est nulle.
En 2013, Marco a introduit la notion d’entropie polynomiale, qui mesure
le taux de croissance polynomial de la complexité du système dynamique, et a démontré que l’entropie polynomiale des fonctions hamiltoniennes de Morse sur une surface symplectique est 0,1 ou 2, cette dernière valeur n’étant atteinte que lorsque la fonction présente des points critiques d’indice 1. Dans mon
exposé, je présente une généralisation de ce résultat à des fonctions
hamiltoniennes “modérément” dégénérées, qui s’affranchit des formes
normales locales au voisinage des singularités.

3 November 2021 (Dauphine, salle A711) : Donato Scarcella

Title: Solutions asymptotiquement quasipériodiques pour des systèmes hamiltoniens dépendant du temps.

Abstract: En 1954, Kolgomorov, avec son travail surprenant, a commencé ce qui allait prendre le nom de la théorie KAM. Ce résultat a été suivi par ceux d’Arnold, Moser et beaucoup d’autres. La théorie KAM montre la persistance de solutions quasipériodiques dans les systèmes hamiltoniens presque intégrables.  Dans cet exposé nous nous intéresserons aux perturbations qui dépendent du temps. Nous analyserons les propriétés des systèmes hamiltoniens qui convergent asymptotiquement dans le futur vers des systèmes hamiltoniens autonomes ayant un tore invariant admettant des solutions quasipériodiques.  Nous verrons sous quelles hypothèses ces types de systèmes admettent des solutions asymptotiquement quasipériodiques. Puis les applications possibles en mécanique céleste à l’étude d’un système planétaire perturbé par une comète venant de et repartant vers l’infini avec une vitesse non nulle, asymptotiquement dans le futur.

27 October 2021 : holidays

20 October 2021 (Jussieu, salle 15-16-413 ) : Shahriar Aslani

Title: Normal form near orbit segments of a Hamiltonian vector field

Abstract: In this talk, I introduce a local normal form near orbit segments of a convex Hamiltonian vector field. One might correspond this normal form to Fermi coordinates in Riemannian geometry. The normal form is applicable in the study of perturbed linearized Poincare maps; in particular, when we perturb a given Hamiltonian by adding a function that only depends on the position variables. That is the so-called Mane perturbation. The concept of Mane perturbation is closely related to conformal perturbation of Riemannian metrics.
Based on bumpy metric theorem, non-degeneracy of all closed geodesics of a Ck Riemannian metric is a generic property with respect to Whitney Ck-topology. To prove a bumpy metric like theorem “à la Mane” for convex Hamiltonian systems, it is crucial to study the effects of Mane perturbations on linearized Poincaré maps.

13 October 2021 (Jussieu, salle 15-16-413) : Simon Allais

Titre : Sur les points translatés des contactomorphismes des espaces lenticulaires

Résumé : En 2011, Sandon montra que les points translatés des contactomorphismes isotopes à l’identité des espaces projectifs réels munis de la forme de contact standard existaient toujours en un nombre supérieur à une quantité liée à la topologie de ces espaces. Elle en conjectura un analogue de la conjecture d’Arnol’d pour les contactomorphismes isotopes à l’identité de variétés de contact quelconques.
Dans cet exposé, nous expliquerons cette conjecture et comment l’usage de fonctions génératrice permet de la démontrer dans les espaces lenticulaires standard.

6 October 2021 (Jussieu, salle 15-16-413 ) : Santiago Barbieri

Titre : FLEXIBILITY AND ANALYTIC SMOOTHING IN AVERAGING THEORY
(joint work with J.P. Marco and J. Massetti)

Résumé : The main goal of this work is to introduce a unified way for proving “long time stability” of the action variables for perturbations of completely integrable Hamiltonian systems which belong to a large class of function spaces. We will limit ourselves here to Hölder perturbations of analytic systems, but our method is flexible enough to be adapted to many other settings. By using techniques of analytic smoothing introduced by Jackson, Moser and Zehnder, together with a new improved estimate on the Fourier norm of the smoothed function, we extend the classical Nekhoroshev’s estimates to the case of Hölder regular perturbations of analytic steep near-integrable hamiltonian systems, the stability times being polynomially long in the inverse of the size of the perturbation. We prove that the stability exponents can be taken to be $a=\frac{\ell-1}{\alpha_1…\alpha_{n-2}}+1/2$ for the time of stability and $b=\frac{1}{2n\alpha_1…\alpha_{n-1}}$ for the radius of stability, $\ell>n + 1$ being the regularity and the $\alpha_i$’s being the indices of steepness. More in detail, Nekhoroshev’s original construction consists of two parts

1) A geometric part, in which a suitable patchwork of the phase space is constructed. The elements of such a covering are “resonant blocks”, i.e. regions in which the frequency vector of the unperturbed hamiltonian satisfies a certain number of resonance relations with respect to certain maximal resonant lattices.

2) An analytic part, in which the hamiltonian is put into resonant normal form. The size of the remainder in the normal form depends on the regularity at hand: it is exponentially small in analytic and Gevrey class and polynomially small in Hölder class.

The steepness condition then assures that the actions are confined over a time which is given by the ratio of the associated analyticity width with the size of the remainder of the normal form. Here, differently from the standard approaches, we do not construct any resonant normal form in Hölder class. Rather, we use analytic smoothing techniques on the perturbation at hand, and we apply the classic analytic normal form directly to the smoothed hamiltonian. This allows for much simpler calculations. Moreover, we find a new estimate on the Fourier norm of the smoothed function that, together with the good choice of the parameters in the resonant patchwork introduced by Guzzo, Chierchia and Benettin, allows us to obtain the exponents of stability $a$ and $b$, which cannot be taken any better by making use of these arguments.